The map \({K}\)  was defined by M. Cahen and S. Gutt in [1]} (see also [2]} or [3]}), where it was shown that \({K}\)  is a moment map for the action of \(\operatorname{Ham}(M, \Omega )\)  on \(\mathbb {S}(M, \Omega )\) . The map \({K}(\nabla )\)  has the form required by Tamarkin's theorem; its part quadratic in the curvature is a constant multiple of the contraction of the first Pontryagin form \(\mathsf {p}_{1}(\nabla )\)  of \(\nabla \)  with \(\Omega _{2}\) : \({K}(\nabla ) = -\delta ^{2}\mathsf {Ric}- (\pi ^{2}/2)\mathsf {p}_{1}(\nabla )_{p}\,^{p}\,_{q}\,^{q}\) . As a consequence, when \(M\)  is compact the integral of \({K}(\nabla )\)  depends only on the cohomology class \([\Omega _{n-2}]\)  of \(\Omega _{n-2}\)  and the first Pontryagin class \([\mathsf {p}_{1}]\)  of \(M\) ,
\(\int _{M}{K}(\nabla )\,\Omega _{n} = -4\pi ^{2}\int _{M} \mathsf {p}_{1}\wedge \Omega _{n-2} = -4\pi ^{2}\langle [\mathsf {p}_{1}] \cup [\Omega _{n-2}], [M]\rangle .\) 
